Method for reducing the calculation time of a numerical calculation for a computer implemented superposition model

ABSTRACT

A method of reducing the computer calculation time of a superposition is disclosed. A computing device having an input unit, an output unit, a memory unit, and an operation unit, is used to calculate the model superposing the function with shifted value of the variable. The model operator is formed by superposing a delta function in the same manner as the superposition of the function. The convolution of a model operator and the function is determined to thereby reduce the calculation time of the model superposing the function with the shifted value of the variable.

BACKGROUND OF THE INVENTION

1. Technical Field

The invention relates to a method of reducing the calculation time for anumerical calculation for a computer implemented superposition model.

2. Discussion of the Prior Art

It is known to compute a superposition model for a waveform functionG(t−τ_(i)), as shown in Formula 1. A sampled arrangement G(I) isinterpolated by shifting by τ_(i) for the respective suffix i to computean arrangement relative to the function G(t−τ_(i)) throughout the domainof the variable t. $\begin{matrix}{{Y(t)} = {\sum\limits_{i = 1}^{N_{i}}{a_{i} \cdot {G\left( {t - \tau_{i}} \right)}}}} & {{FORMULA}\quad I}\end{matrix}$

where:

Y(t): Superposed waveform where (t) time represents a variable, forexample, seismic waves, sound waves, light waves, electromagnetic waves,temperature, voltage, electric current, fluid pressure; additionally,position, displacement, temperature, voltage, electric flow, fluidpressure, or other physical quantities can be substituted for the time.

G(t): Waveform where (t) time represents a variable, for example,seismic waves, sound waves, light waves, electromagnetic waves,temperature, voltage, electric current, fluid pressure; additionally,record of position, displacement, temperature, voltage, electric flow,fluid pressure, or other physical quantities can be substituted for thetime.

a: Coefficient

i: Suffix

τ: Time lag

t: Time

Ni: Number of superposition

{dot over ( )}: Product

The above-described method has some shortcomings. For example:

The function G(t) is a waveform (see FIG. 1) comprising the time t as avariable. The function G(t), however, must first be digitized to beprocessed by a digital computer. That is, data are processed in thecomputer in accordance with the arrangement G(I) sampled at Δt, not thefunction G(t). Accordingly, the function G(t−τ_(i)) must be determinedby interpolation, e.g. by shifting the arrangement G(I) by τ_(i). Thisinterpolation requires significant processing time because therespective time t, i.e. all arrangements, must be interpolated.

Further, the function G(t−τ_(i)) with regard to the respective suffix i,must be multiplied by a coefficient ai. For the purpose of simplifyingthe discussion herein, the arrangement of the function G(t−τ_(i))comprising τ_(i) is interpolated by shifting the arrangement G(I) of thefunction G(t) for τ_(i). All arrangements hereinafter are shown in thismanner. The function G(t−τ_(i)) is treated as a function in amathematical formula, while it is also treated as an arrangement in acalculation. To simplify the discussion herein, arrangements are alsodescribed as functions in some cases. As discussed above, significantprocessing time is required to determine the function arrangement forthe respective suffix i. Therefore, this type of calculation is notpractically implemented in a personal computer.

A superposition model may be used, for example, for the prediction oflarge seismic waves from data describing small seismic waves. FIG. 2illustrates a method of predicting an earthquake at the observationpoints OP. When the waveform G(t) of a small dislocation earthquake SEis observed, displacement occurred on the starting point SP aretransferred in different directions with a time lag τ, finally reachingan arrival point AP. This results in a large earthquake LE in whichdisplacement is amplified over a vast range. See Eiji Kojima, ASemi-Empirical Method for Synthesizing Intermediate-Period Strong GroundMotions, Doctoral Thesis, Tohoku University (August 1996).

In the foregoing prediction model, the delay is τlm, which is thetraveling time of the seismic waves from the small area (l, m) to theobservation point OP, provided that the displacement area is divided insmall pieces and that the seismic waves occur at a respective small area(l, m) with a defined delay.

Formula 2, may be used to predict the large seismic wave Y(t), takinginto consideration the distance decrement, radioactive characteristics,and slip conditions at the respective area (l, m). A waveform of a smallseismic wave G(t) may therefore be processed for superposition topredict a large seismic wave Y(t). $\begin{matrix}{{Y(t)} = {\sum\limits_{l = 1}^{N_{1}}\quad {\sum\limits_{m = 1}^{N_{m}}\quad {\sum\limits_{k = 1}^{N_{k}}\quad {\left( {X_{lomo}/X_{l\quad m}} \right) \cdot \left( {R_{l\quad m}/R_{lomo}} \right) \cdot {G\left( {t - \tau_{l\quad m} - {\left( {k - 1} \right) \cdot \psi}} \right)}}}}}} & {{FORMULA}\quad 2}\end{matrix}$

Where:

Y(t): Large seismic wave representing a ground motion in a largeearthquake.

G(t): Small seismic wave representing a ground motion in a smallearthquake.

X_(lm): Distance from the observation point to the small area created bydividing the displacement of a large earthquake.

X_(lomo): Distance from the seismic center of the small earthquake.

R_(lm): Radioactive characteristic to the observation point of the smallarea created by dividing the displacement of a large earthquake.

R_(lomo): Radioactive characteristic to the observation point of a smallearthquake.

τ: Time lag

t: Time

l, m, k: Suffix

Nl, Nm, Nk: Number of superposition ψ: Time lag of slippage in a smallearthquake.

SUMMARY OF THE INVENTION

This invention resolves the above disadvantages of the prior arttechniques by providing a method for reducing the calculation time forthe numerical calculation of a computer implemented superposition model.

In a first embodiment of the invention, a method is provided forreducing the calculation time of a numerical calculation for a computerimplemented superposition model. This method includes a step forcalculating a model of superposing the function with shifted values fora variable using a computer equipped with an input unit, an output unit,and a memory unit. A model operator is formed by superposing the deltafunction in the same manner as the superposed function model todetermine the composition product of the model operator and thefunction.

In a second embodiment of the invention, a method is provided forreducing the calculation time of a numerical calculation for a computerimplemented superposition model, in which the model is computed pluraltimes.

In a third embodiment of this invention, a method is provided forreducing the calculation time for a numerical calculation for a computerimplemented superposition function model, in which a variable of thefunction is time, location, temperature, or other physical quantities.

In a fourth embodiment of this invention, a medium is provided for useby computer and on which a program is stored for reducing thecalculation time of a computer for the numerical calculation of asuperposition model. Using the program stored on the medium, a model ofsuperposing the function having shifted values for a variable iscalculated by a computer equipped with an input unit, an output unit,and a memory unit. A model operator is formed by superposing a deltafunction in the same manner as the superposed function model todetermine the composition product of the model operator and thefunction.

In a fifth embodiment of the invention, a medium is provided for use bya computer and having therein a program for reducing the calculationtime of a numerical calculation of a superposition, the model iscomputed plural times.

In a sixth embodiment of the invention, a medium is provided for storinga program for reducing the calculation time of a numerical calculationof a superposition model, a variable of the function is time, location,temperature, or other physical quantities.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an arrangement of the function G(I) relating to asmall earthquake;

FIG. 2 illustrates a model of predicting a waveform of a largeearthquake resulting from a small earthquake;

FIG. 3 is an approximate curved line of the delta function;

FIG. 4 is another approximate curved line of the delta function;

FIG. 5 is a flowchart of the computer processing for Formula 4;

FIG. 6 illustrates a calculation of the interpolation of the deltafunction;

FIG. 7 is a flowchart of the computer processing for Formula 3;

FIG. 8 illustrates an arrangement of the model operator F(I) forpredicting a small earthquake; and

FIG. 9 illustrates an arrangement of the function Y(I) relating to alarge earthquake.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

In forming various models such as a prediction model or a control model,functions relating to such factors as prediction or control aredetermined by superposing the functions necessary for such prediction orcontrol based upon the production models.

For example, provided that the model simply has only one Σ for thesummation of the waveform in a small earthquake, the waveform may berepresented as the function G(t), and waveforms of a large earthquakemay be predicted by superposing the waveforms and by the function Y(t)of Formula 3 herein. Here, the reference signs t and τare used in anexample of the waveform in an earthquake, thereby representing time andthe time lag, respectively. In other examples, the function G(t) may beother physical quantities such as sound-waves, light waves,electromagnetic waves, temperature, voltage, electric current, or fluidpressure. The variable t may also be other physical quantities, such aspositions, displacement, temperature, electric current, voltage, orfluid pressure. $\begin{matrix}{{Y(t)} = {\sum\limits_{i = 1}^{N_{i}}{a_{i} \cdot {G\left( {t - \tau_{i}} \right)}}}} & {{FORMULA}\quad 3}\end{matrix}$

where:

Y(t): Superposed wave where (t) time represents a variable, for example,seismic waves, sound waves, light waves, or electromagnetic waves.

G(t): Waveform where (t) time represents a variable, for example,seismic waves, sound waves, light waves, or electromagnetic waves.

In Formula 3, the model of superposing the function G(t) may be replacedby Formula 4 and Formula 5, where Formula 4 is composed of theconvolution of the model operator F(t) and the function G(t). and whereFormula 5 is composite integration of the Dirac's delta function. Afterthe replacement, the variable (t−τ_(i)) shifts from the function G(t−τ_(i)) to the Dirac's delta function δ (t−τ_(i)). Then, thesuperposing operation of the function G(t−τ_(i)) shifts to the modeloperator F(t). The model operator F(t) is the superposition of the deltafunction δ (t−τ_(i)).

Y(t)=F(t)*G(t)  FORMULA 4

where:

F(t): Model operator of superposed dislocation

*: Convolution $\begin{matrix}{{F(t)} = {\sum\limits_{i = 1}^{N_{i}}\quad {a_{i} \cdot {\delta \left( {t - \tau_{i}} \right)}}}} & {{FORMULA}\quad 5}\end{matrix}$

where:

δ(t): Dirac's delta function

The delta function δ (t−τ_(i)) becomes zero at points other than thepoint where the variable t is in the neighborhood of τ_(i) and becomeszero at most domains of the variable t, thereby enabling straightforwardcalculation of the delta function of the respective suffix.

In computing the arrangement of the model operator F(t) in which thedelta function δ (t−τ_(i)) is superposed, τ_(i) is included in a groupwithin the same sampling time Δt, and the wave of the model operatorF(t) becomes zero at most domains. Thus, this process makes it possibleto increase the processing speed for the convolution of the modeloperator F(t) and the function G(t).

When the function G(t) is a physical phenomenon, τ_(i) included in agroup within the same sampling time Δt at many suffixes because of, e.g.the relativity. For example, if an earthquake has a far fieldobservation point, as shown in FIG. 2, the time lag τ_(i) ofsuperposition in the transmitting period of the displacement and that ofthe seismic wave has almost the same value at the concentric circle ofthe starting point SP, and is therefore in the group within the samplingtime Δt. Accordingly, the arrangement of the model operator F(t) becomessmaller in comparison to the number of the suffix i, thereby reducingthe calculation time of the convolution of Formula 4.

The delta function δ (t) is defined in Formula 6 below. The deltafunction may be generalized as in Formula 7 below. The function dΔt(t)of Formula 7 has numerous functions. The example of the delta functionis shown in FIG. 3, where the value of Δt in the delta function is verysmall. It is 1/Δt when the variable t is zero, and the value of Δt iszero when the variable t is −Δt or smaller, or when the variable t is+Δt or larger. Thus, it is possible to express a straight line between−Δt and Δt as an approximate formula. This approximate formula isexplained in Formula 8 below. Δt this point, the delta functionδ(t−τ_(i)) may be determined by a straight line interpolation of thesuffix i. For example, as shown in FIG. 4, the value of Δt in otherexamples of the delta function is very small, and when the variable t isbetween −Δt/2 and Δt/2, the value is 1/Δt. When variable t is −Δt/2 orsmaller, or when the variable t is Δt/2 or larger, the value becomeszero. Formula 9 below provides an approximate formula of this example.

In FIG. 4, the delta function δ (t) is equal to 1/Δt when t is equal to−Δt/2 where the formula δ(t)=1/Δt is represented by a black dot. Thedelta function δ (t) is equal to zero when t is equal to Δt/2, where theformula δ(t)=0 is represented by a black dot. Alternatively, byswitching the black and white dots, the delta function δ(t) is equal tozero when t is equal to −Δt/2, and the delta function b(t) is equal to1/Δt when t is equal to Δt/2.

f(t)=f(t)*δ(t)  FORMULA 6

f(t): Continuously differentiable function and

limf(t)=0, limf(t)=0 simultaneously t→∞ t→∞−→∞

δ(t): Dirac's delta function $\begin{matrix}{{\delta (t)} = {\lim\limits_{{\Delta \quad t}\rightarrow 0}{d_{\Delta \quad t}(t)}}} & {{FORMULA}\quad 7}\end{matrix}$

where:

d_(Δt)(t): Modeled and discreted delta function

Δt: Infinitesimal value (Δt may be other than a sampling time)

When|t|≧Δt, d _(Δt)(t)=0

When|t|<Δt, d _(Δt)(t)=1/Δt−|t|/(Δt)²  FORMULA 8

where:

d_(Δt)(t): Modeled and discreted delta function

t: Time

Δt: Sampling time

When t<−Δt/2, Δt/2≦t, d _(Δt)(t)=0

When−Δt/2≦t<Δt/2, d _(Δt)(t)=1  FORMULA9

The model operator F(t) is shown in Formula 5, where the arrangement ofthe delta function is multiplied by the arrangement of the coefficienta. Computing the superposition of the model operator F(t), for example,may be done by the use of a computer following the process of FIG. 5.The flowchart shown in FIG. 5 is an example of the simple explanation ofthe process, and it is not designed to provide a calculation forprogramming or increasing the computing speed.

In calculating the model operator of FIG. 5, the function in FIG. 3 maybe applied as in FIG. 6. The delta function is shifted for the time lagτ_(i), the arrangement TAU(3), to determine the digitized value of thedelta function at every time. In FIG. 6, when Δt is 0.01 second and thetime lag τ3 is 0.0375 second, the model operator F(t) determines thevalues of the arrangement F(4) and the arrangement F(5). The otherarrangement F(I) becomes zero. When the time lag τ8 is 0.0310, thefunction F(t) determines the values of the arrangement F(4) and thearrangement F(5) as when the time lag τ3 is 0.0375. The otherarrangement F(I) becomes zero. Even when the respective time lag is τ3and τ8, each may be superposed at the common location I of thearrangement F(I). The arrangement F(I) is determined by repeating thecomputation of all suffixes. A flowchart of FIG. 7 shows the process ofcomputing the arrangement of the convolution Y(t) of the arrangement ofthe function G(t) by use of the model operator F(t).

FIG. 7 is a flow diagram that illustrates a process for calculating thevalue of the convolution Y(t) at every sampling time Δt. The modeloperator F(t) and the function G(t) are shifted every Δt to multiply oneanother to determine the total convolution Y(t) in addition to theconvolution Y(t) at the certain sampling time Δt. Accordingly, thearrangement of the convolution Y(t) is determined.

The calculation time may be reduced in the same way where a pluralnumber of summations exist, as shown in Formula 2, i.e. a plural numberof summation signs Σ exist. For example, the superposition formula ofFormula 2 may be adapted as shown by the following Formulae 10-12.Formulae 10-12 increase the computing speed just as Formulae 4 and 5.

 Y(t)=E(t)*F(t)*G(t)  FORMULA 10

where:

Y(t): Superposed waveform where (t) time represents a variable

E(t): Model operator for superposed dislocation

F(t): Model operator for superposed slippage

G(t): Waveform where (t) time represents a variable $\begin{matrix}{{E(t)} = {\sum\limits_{l = 1}^{N_{1}}\quad {\sum\limits_{m = 1}^{N_{m}}\quad {\left( {X_{lomo}/X_{l\quad m}} \right) \cdot \left( {R_{l\quad m}/R_{lomo}} \right) \cdot {\delta \left( {t - \tau_{l\quad m}} \right)}}}}} & {{FORMULA}\quad 11}\end{matrix}$

where:

E(t): Model operator for superposed dislocation

δ(t): Dirac's delta function $\begin{matrix}{{F(t)} = {\sum\limits_{k = 1}^{N_{k}}\quad {\delta \left( {t - {\left( {k - 1} \right) \cdot \psi}} \right)}}} & {{FORMULA}\quad 12}\end{matrix}$

where:

F(t): Model operator for superposed slippage

δ(t): Dirac's delta fiction

The same process may be used to reduce the calculation time indetermining the summation of the function with the plural number ofdifferential as in Formula 13. For instance, in this particularsituation, Formula 13 may be substituted for Formula 3. Formula 14increases the calculation speed in the same process explained in Formula4. $\begin{matrix}{{Y(t)} = {{\sum\limits_{i = 1}^{N_{1}}\quad {a_{i1} \cdot {G_{1}\left( {t - \tau_{i}} \right)}}} + {\sum\limits_{i = 1}^{N_{2}}\quad {a_{i2} \cdot {{\overset{.}{G}}_{2}\left( {t - \tau_{i}} \right)}}} + {\sum\limits_{i = 1}^{N_{3}}\quad {a_{i3} \cdot {{\overset{¨}{G}}_{3}\left( {t - \tau_{i}} \right)}}} + \ldots}} & {{FORMULA}\quad 13}\end{matrix}$

where:

Y(t): Superposed waveform where (t) time represents a variable, forexample, seismic waves, sound waves, light waves, electromagnetic waves,voltage for control, or electric current for control.

G₁(t), G₂(t), G₃(t) . . . : Waveform, for example, seismic waves, soundwaves, light waves, electromagnetic waves, electric current, or voltage.

a_(ij): Coefficient

N₁, N₂, N₃ . . . : Numbers

{dot over ( )}: First differential of t

{umlaut over ( )}: Second differential of t

Y(t)=F ₁(t)*G ₁(t)+F ₂(t)*{dot over (G)} ₂(t)+F ₃(t)*{umlaut over (G)}₃(t)+  FORMULA 14

where:

Y(t): Superposed waveform where (t) time represents a variable, forexample, seismic waves, sound waves, light waves, electromagnetic waves,voltage for control, or electric current for control.

G₁(t), G₂(t), G₃(t) . . . : Waveform, for example, seismic waves, soundwaves, light waves, electromagnetic waves, electric current, or voltage.

F₁, F₂ . . . : Model operator

{dot over ( )}: First differential of t

{umlaut over ( )}: Second differential of t

Provided that G₁(t), G₂(t), G₃(t) . . . are same in both Formula 13 andFormula 14, Formula 16 is determined by applying Formula 15, which isthe differential with the delta function, and by defining the new modeloperator of Formula 17. Function 16 simplifies the calculation enablinga further reduction of the calculation time of the computer. In Formulae13-17, the differential may be replaced with an integration.

{dot over (f)}(t)=δ(t)*{dot over (f)}(t)={dot over(δ)}(t)*f(t)  FORMULA15

where:

{dot over ( )}: First differential of t

f(t): Continuously differentiable function and limf(t)=0, limf(t)=0simultaneously t→∞ t→−∞

Y(t)=F ₁(t)*G(t)+F ₂(t)*{dot over (G)}(t)+F ₃(t)*{umlaut over (G)}(t)+ .. . ={F ₁(t)+{dot over (F)} ₂(t)+{umlaut over (F)} ₃(t)+ . . .}*G(t)=F(t)*G(t)  FORMULA 16

where:

Y(t):. Superposed waveform where (t) time represents a variable, forexample, seismic waves, sound waves, light waves, electromagnetic waves,voltage for control, or electric current for control.

G(t): Waveform, for example, seismic waves, sound waves, light waves, orelectromagnetic waves.

F₁(t), F₂(t) . . . : Model operator

{dot over ( )}: First differential of t

{umlaut over ( )}: Second differential of t

F(t)=F ₁(t)+F ₂(t)+F ₃(t)+  FORMULA 17

where:

F(t): New model operator

F₁(t), F₂(t) . . . : Model operator

Calculation of the superposition model, e.g. the calculation of thesuperposition model shown in the flow charts of FIG. 5 and FIG. 7, isprocessed with a program saved in a computer language. This program maybe saved on a medium such as a floppy disk, a hard disk, or a memory,and a computer reads the program on the medium to superpose when thecalculation is necessary.

The following is an example of one application of this invention whenpredicting a large earthquake from a small earthquake.

The relation of Formula 3 is determined through a simplified model forpredicting the waveform of a large earthquake from the waveform of asmall earthquake. The coefficient a of Formula 3 is the correctioncoefficient for creating an assumption that the displacement started atthe starting point to extend to the certain point, the suffix i, wherethe small earthquake occurred. The time lag τ_(i) relates to the perioduntil the small earthquake occurs at the destination point, the suffixi, and is observed from the beginning of the displacement. The functionG(t−τ_(i)) is the waveform of the small earthquake predicted at theobservation point, the suffix i. The function Y(t) is a waveform of thepredictable large earthquake.

Formula 3 is modified to form Formula 4 and Formula 5, and the modeloperator F(t) is determined by the calculation of Formula 5 followingthe flowchart of FIG. 5. This arrangement is shown in FIG. 8. Thearrangement in FIG. 8 is focused on a certain time, which decreases thenumber of the value for an arrangement that is not zero. The modeloperator F(t) is simplified and Formula 4 is simplified as well toreduce the calculation time of the waveform Y(t) of a large earthquakethat is predictable by the numerical calculation.

FIG. 1 (discussed above) illustrates an arrangement of the function G(t)with respect to a small earthquake. At this point, the arrangement ofthe model operator F(t) determined above and the convolution of thearrangement of the function G(t) are computed in accordance with theprocess shown in of FIG. 7. FIG. 9 shows the result of the calculation.During the calculation, the calculation time of the convolution for thearrangement of the model operator F(t) is dramatically reduced becausethe number of the arrangement is very small.

In Formula 4, the model operator F(t) is predetermined by the predictionmodel, and the function G(t) is the seismic waves in a small area, whichis subject to the real time observation. Formula 4 is the compositeintegration of F(t) and G(t). Therefore, the observed portion of G(t) iscomposited in F(t) whenever G(t) is observed, and the result of thisoperation is added to the previously determined result which wasdetermined through the above-described same operation, thereby makingthe real time calculation of the function Y(t) possible. Accordingly,the real time calculation becomes possible by high speed numericalprocessing by personal computers.

The operation, i.e. real time calculation is not possible when thevariable in the operation is the frequency w; for observing the waveformof the whole area, however, if the variable is time t, a real timecalculation is possible. Using the waveform during the observation (ifthe waveform not observed is assumed to be zero), G(t) requires afrequency analysis every time, which slows down the computation on acomputer.

The invention is not limited to the embodiments described above. Forexample, real time control is applicable to a vibration control methodfor architecture, a real time positioning control mechanism with GPS, afactory control system, a control mechanism for spraying concrete on arough surface, a robot control mechanism, or a quality control mechanismfor merchandise. If the above-explained superposition control model isapplicable, real time control is possible in that mechanism.

The invention provides at least one of the following advantages. Forexample, the model operator reduces the computer calculation time for asuperposed function model. When the function indicates physicalphenomena, the relativity reduces computer calculation time for thesuperposed function model. The model operator is used to replace Formula3 with the Formula 4 and Formula 5, which minimizes the chances of acalculation differences. A personal computer implementing the inventionmay be used for heavy computation. Thus, the invention provides easyaccess by use of a computer to high-level computing mechanisms for aconsiderably lower cost. This invention also eliminates the situationthat a computer operator is required to occupy the computer for quite along period of time due to the significant processing time required.

It is readily apparent that the above-described has the advantage ofwide commercial utility. It should be understood that the specific formof the invention hereinabove described is intended to be representativeonly, as certain modifications within the scope of these teachings willbe apparent to those skilled in the art. Accordingly, reference shouldbe made to the following claims in determining the full scope of theinvention.

What is claimed is:
 1. A method of predicting movement of matter orenergy by calculating a discrete superposition model using a computerterminal, the method comprising the steps of: replacing a firstsuperposition model represented by the following formula:${Y(t)} = {\sum\limits_{i = 1}^{N_{i}}{a_{i}^{*}{G\left( {t - \tau_{i}} \right)}}}$

with a second superposition model represented by the following formula:${{Y(t)} = {\left\{ {\sum\limits_{i = 1}^{N_{i}}{a_{i}^{*}{\delta \left( {t - \tau_{i}} \right)}}} \right\}*{G(t)}}};\quad \text{and}$

calculating on the computer terminal a discrete version of the secondsuperposition model interpolating only a delta function, wherein Y(t) isa superposed wave; t represents time; N_(i) is a selected number ofsuperpositions; a_(i) is a coefficient; i is a suffix; τ is time lag;δ(t−τ_(i)) is the delta function; * is convolution; G(t) is a sampledwaveform without the variable time lag; and wherein the delta functionδ(t−τ_(i)) is zero other than at points where the variable t is within aselected range of τ thereby reducing the computer processing time. 2.The method of claim 1, wherein said model is computed plural times.
 3. Amedium for storing a program of predicting movement of matter or energyby calculating a discrete superposition model using a computer terminal,the method comprising the steps of: replacing a first superpositionmodel represented by the following formula:${Y(t)} = {\sum\limits_{i = 1}^{N_{i}}{a_{i}^{*}{G\left( {t - \tau_{i}} \right)}}}$

with a second superposition model represented by the following formula:${{Y(t)} = {\left\{ {\sum\limits_{i = 1}^{N_{i}}{a_{i}^{*}{\delta \left( {t - \tau_{i}} \right)}}} \right\}*{G(t)}}};\quad \text{and}$

calculating on the computer terminal a discrete version of the secondsuperposition model interpolating only a delta function, wherein Y(t) isa superposed wave; t represents time; N_(i) is a selected number ofsuperpositions; a_(i) is a coefficient; i is a suffix; τ is time lag;δ(t−τ_(i)) is the delta function; * is convolution; G(t) is a sampledwaveform without the variable time lag; and wherein the delta functionδ(t−τ_(i)) is zero other than at points where the variable t is within aselected range of τ thereby reducing the computer processing time. 4.The medium of claim 3, wherein said model is computed plural times.